The asymptotic representation of some series and the Riemann hypothesis (original) (raw)
Mathematics and Statistics, 2022
The Riemann zeta (ζ) function ζ(s) = ∞ n=1 1 n s is valid for all complex number s = x + iy : Re(s) > 1, for the line x = 1. Euler-Riemann found that the function equals zero for all negative even integers: −2, −4, −6, • • • (commonly known as trivial zeros) has an infinite number of zeros in the critical strip of complex numbers between the lines x = 0 and x = 1. Moreover, it was well known to him that all non-trivial zeros are exhibiting symmetry with respect to the critical line x = 1 2. As a result, Riemann conjectured that all of the non-trivial zeros are on the critical line, this hypothesis is known as the Riemann hypothesis. The Riemann zeta function plays a momentous part while analyzing the number theory and has applications in applied statistics, probability theory and Physics. The Riemann zeta function is closely related to one of the most challenging unsolved problems in mathematics (the Riemann hypothesis) which has been classified as the 8th of Hilbert's 23 problems. This function is useful in number theory for investigating the anomalous behavior of prime numbers. If this theory is proven to be correct, it means we will be able to know the sequential order of the prime numbers. Numerous approaches have been applied towards the solution of this problem, which includes both numerical and geometrical approaches, also the Taylor series of the Riemann zeta function, and the asymptotic properties of its coefficients. Despite the fact that there are around 10 13 , non-trivial zeros on the critical line, we cannot assume that the Riemann Hypothesis (RH) is necessarily true unless a lucid proof is provided. Indeed, there are differing viewpoints not only on the Riemann Hypothesis's reliability, but also on certain basic conclusions see for example [16] in which the author justifies the location of non-trivial zero subject to the simultaneous occurrence of ζ(s) = ζ(1 − s) = 0, and omitting the impact of an indeterminate form ∞.0, that appears in Riemann's approach. In this study we also consider the simultaneous occurrence ζ(s) = ζ(1 − s) = 0 but we adopt an element-wise approach of the Taylor series by expanding n −x for all n = 1, 2, 3, • • • at the real parts of the non-trivial zeta zeros lying in the critical strip for s = α + iy is a non-trivial zero of ζ(s), we first expand each term n −x at α then at 1 − α. Then In this sequel, we evoke the simultaneous occurrence of the non-trivial zeta function zeros ζ(s) = ζ(1 − s) = 0, on the critical strip by the means of different representations of Zeta function. Consequently, proves that Riemann Hypothesis is likely to be true.
Towards A Solution of The Riemann Hypothesis
In 1859, Georg Friedrich Bernhard Riemann had announced the following conjecture, called Riemann Hypothesis : The nontrivial roots (zeros) s=sigma+its=\sigma+its=sigma+it of the zeta function, defined by: \zeta(s) = \sum_{n=1}^{+\infty}\frac{1}{n^s},\,\mbox{for}\quad \Re(s)>1$$ have real part sigma=1/2\sigma=1/2 sigma=1/2}. We give a proof that sigma=1/2\sigma= 1/2 sigma=1/2 using an equivalent statement of the Riemann Hypothesis concerning the Dirichlet eta\etaeta function.
On A Rapidly Converging Series For The Riemann Zeta Function
JP Journal of Algebra, Number Theory and Applications
To evaluate Riemann's zeta function is important for many investigations related to the area of number theory, and to have quickly converging series at hand in particular. We investigate a class of summation formulae and find, as a special case, a new proof of a rapidly converging series for the Riemann zeta function. The series converges in the entire complex plane, its rate of convergence being significantly faster than comparable representations, and so is a useful basis for evaluation algorithms. The evaluation of corresponding coefficients is not problematic, and precise convergence rates are elaborated in detail. The globally converging series obtained allow to reduce Riemann's hypothesis to similar properties on polynomials. And interestingly, Laguerre's polynomials form a kind of leitmotif through all sections.
Some Remarks and Propositions on Riemann Hypothesis
Mathematics and Statistics, 2021
In 1859, Bernhard Riemann, a German mathematician, published a paper to the Berlin Academy that would change mathematics forever. The mystery of prime numbers was the focus. At the core of the presentation was indeed a concept that had not yet been proven by Riemann, one that to this day baffles mathematicians. The way we do business could have been changed if the Riemann hypothesis holds true, which is because prime numbers are the key element for banking and e-commerce security. It will also have a significant influence, impacting quantum mechanics, chaos theory, and the future of computation, on the cutting edge of science. In this article, we look at some well-known results of Riemann Zeta function in a different light. We explore the proofs of Zeta integral Representation, Analytic continuity and the first functional equation. Initially, we observe omitting a logical undefined term in the integral representation of Zeta function by the means of Gamma function. For that we propound some modifications in order to reasonably justify the location of the non-trivial zeros on the critical line: = 1 2 by assuming that () and (1 −) simultaneously equal zero. Consequently, we conditionally prove Riemann Hypothesis. MSC 2010 Classification: 97I80, 11M41
viXra, 2018
This paper explicates the Riemann hypothesis and proves its validity; it explains why the non-trivial zeros of the Riemann zeta function ζ will always be on the critical line Re(s) = 1/2 and not anywhere else on the critical strip bounded by Re(s) = 0 and Re(s) = 1. Much exact calculations are presented, instead of approximations, for the sake of accuracy or precision, clarity and rigor. (N.B.: New materials have been added to the paper.
On Non-Trivial Zeros and Riemann Zeta Function
This paper examines the mysterious non-trivial zeros of the Riemann zeta function ζ and explains their role, e.g., in the computation of the error term in Riemann’s J function for estimating the quantity of primes less than a given number. The paper also explains the close connection between the Riemann zeta function ζ and the prime numbers. [Published in international mathematics journal.]
The Non-Trivial Zeros Of The Riemann Zeta Function
This paper shows why the non-trivial zeros of the Riemann zeta function ζ will always be on the critical line Re(s) = 1/2 and not anywhere else on the critical strip bounded by Re(s) = 0 and Re(s) = 1, thus affirming the validity of the Riemann hypothesis. [The paper is published in a journal of number theory.]
A simple proof of the Riemann Hypothesis
We show that the Riemann zeta function ζ(s) does not have any zeroes for Res > 1/2 and therefore does not have any zeroes for 0 < Res < 1/2. In order to do that, we rewrite the Riemann zeta function as a product of the related fuctions ζ_{2^i , j} (s). We show that for Res > 1/2 each of these fuctions is zero only if ζ(s) is zero. Therefore we derive that for Res > 1/2 each zero of ζ(s) is of the degree n , where n can be arbitrary. It follows that there are no zeros of ζ(s) for Res > 1/2. This paper with the proof as it is presented here was written on 10/25/2007. Lemmas 1,2,3 had been added recently to explain how to obtain the estimates (10) and (11) which are important in the proof. Lemma 4 had been added to explain how to construct the analytic continuation of the functions ζ_{2^i , j} (s) to Res > 1/2 .